Mathematical Induction - Discrete Structures - Lecture Slides, Slides of Discrete Structures and Graph Theory

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• Mathematical

• Induction

• The principle of mathematical induction is a useful tool for proving that a certain predicate is true for all natural numbers.
• It cannot be used to discover theorems, but only to prove them.

• Example:
• Show that n < 2n^ for all positive integers n.
• Let P(n) be the proposition “n < 2n^ .”
• 1. Show that P(1) is true. (basis step)
• P(1) is true, because 1 < 2^1 = 2.

• 2. Show that if P(n) is true, then P(n + 1) is true. (inductive step)
• Assume that n < 2n^ is true.
• We need to show that P(n + 1) is true, i.e.
• n + 1 < 2n+
• We start from n < 2n^ :
• n + 1 < 2n^ + 1 ≤ 2 n^ + 2 n^ = 2n+
• Therefore, if n < 2n^ then n + 1 < 2n+

• Another Example (“Gauss”):
• 1 + 2 + … + n = n (n + 1)/

1. Show that P(0) is true. (basis step)

• For n = 0 we get 0 = 0. True.

2. Show that if P(n) then P(n + 1) for any n∈N. (inductive step)

• 1 + 2 + … + n = n (n + 1)/
• 1 + 2 + … + n + (n + 1) = n (n + 1)/2 + (n + 1)
• = (2n + 2 + n (n + 1))/
• = (2n + 2 + n^2 + n)/
• = (2 + 3n + n^2 )/
• = (n + 1) (n + 2)/
• = (n + 1) ((n + 1) + 1)/

• There is another proof technique that is very similar to the principle of mathematical induction.
• It is called the second principle of mathematical induction.
• It can be used to prove that a propositional function P(n) is true for any natural number n.

• The second principle of mathematical induction:
• Show that P(0) is true. (basis step)
• Show that if P(0) and P(1) and … and P(n), then P(n + 1) for any n∈N. (inductive step)
• Then P(n) must be true for any n∈N. (conclusion)

• Show that if P(2) and P(3) and … and P(n), then P(n + 1) for any n∈N. (inductive step)
• Two possible cases:
• If (n + 1) is prime , then obviously P(n + 1) is true.
• If (n + 1) is composite , it can be written as the product of two integers a and b such that 2 ≤ a ≤ b < n + 1.
• By the induction hypothesis , both a and b can be written as the product of primes.
• Therefore, n + 1 = a⋅b can be written as the product of primes.

• Then P(n) must be true for any n∈N. (conclusion)
• End of proof.
• We have shown that every integer greater than 1 can be written as the product of primes.